11 Febbraio, 2013 11:00 oclock

Stokes and Navier-Stokes equations under slip or leak boundary conditions of friction type

Takahito Kashiwabara, University of Tokyo

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

We consider the incompressible flow with some
nonlinear boundary conditions, called the slip (resp. leak) boundary condition of friction type.
They basically mean that: if the tangential (resp. normal) component of the fluid stress remains under the given threshold, there occurs no tangential (resp. normal) flow i.e. slip (resp. leak); otherwise slip (resp. leak) is allowed to take place on the boundary.
In other words, they are nonlinear intermediates between Dirichlet and Neumann boundary conditions. We can also regard them as fluid-dynamical versions of the Coulomb-type friction condition known in elasticity.

This talk presents, from a theoretical viewpoint, numerical and analytical studies for those boundary conditions. First we notice that a weak form of this problem is written by a variational inequality of the second kind. There appears an L1 norm defined on the boundary, which reflects the nonlinearity of the boundary conditions.
Then, in the numerical part, we propose a discretization by FEM for the stationary Stokes case, discussing error analysis and solution algorithm.

The key point is the introduction of numerical-integration formulas for the L1 norm.
This enables us to preserve in a discrete sense some nice properties known in the continuous (i.e. PDE) level.
For example, the continuous variational inequality problem is known to admit an equivalent saddle point formulation, and so does our discrete variational inequality.
Finally, in the analytical part, the nonstationary Navier-Stokes case is addressed, where we prove the existence and uniqueness of a strong solution. The technique is rather standard and classical; we regularize the L1-norm term to reduce the variational inequality into an equation, construct Galerkin approximations, derive a priori estimates of Ladyzhenskaya type, and pass to the limit.

What is not standard is a treatment of the compatibility condition at the initial condition, where we need to perturb the initial velocity.
Another remark is that in the case of leak boundary condition, we cannot obtain a global-in-time solution without assuming some smallness on data, even in the 2D case.